Energy Transfer in Master
Equation Simulations:
A New Approach
JOHN R. BARKER
Department of Atmospheric, Oceanic, and Space Sciences, University of Michigan, Ann Arbor, MI 48109-2143
Received 5 February 2009; revised 14 April 2009, 19 June 2009; accepted 22 June 2009
DOI 10.1002/kin.20447
Published online in Wiley InterScience (www.interscience.wiley.com).
ABSTRACT: Collisional energy transfer plays a key role in recombination, unimolecular, and
chemical activation reactions. For master equation simulations of such reaction systems, it
is conventionally assumed that the rate constant for inelastic energy transfer collisions is
independent of the excitation energy. However, numerical instabilities and nonphysical results
are encountered when normalizing the collision step-size distribution in the sparse density
of states regime at low energies. It is argued here that the conventional assumption is not
correct, and it is shown that the numerical problems and nonphysical results are eliminated
by making a plausible assumption about the energy dependence of the rate coefficient for
inelastic collisions. The new assumption produces a model that is more physically realistic for
any reasonable choice of collision step-size distribution, but more work remains to be done. The
resulting numerical algorithm is stable and noniterative. Testing shows that overall accuracy in
master equation simulations is better with this new approach than with the conventional one.
C 2009
This new approach is appropriate for all energy-grained master equation formulations. Wiley Periodicals, Inc. Int J Chem Kinet 41: 748–763, 2009
INTRODUCTION
Collisional energy transfer is a key feature of most
chemical reaction systems. Collisions activate and deactivate molecular species while establishing the Boltzmann energy distribution. At low pressures, collisional
energy transfer is slow and therefore controls the rate of
unimolecular reactions. The collision–reaction master
equation has proved to be very effective in describing unimolecular reactions in their many variations
[1–3], which are categorized according to how the reaction is initiated: thermal activation, photoactivation,
Correspondence to: John R. Barker; e-mail: jrbarker@umich.
edu.
Contract grant sponsor: NASA (Planetary Atmospheres).
Contract grant sponsor: NSF (Atmospheric Chemistry).
c 2009 Wiley Periodicals, Inc.
chemical activation, etc. In all of these, collisional
energy transfer plays an important role.
As discussed elsewhere [4,5], energy transfer involving polyatomic molecules was first investigated as
a part of unimolecular reaction studies, but was later
investigated in nonreactive systems by “direct” timeresolved experimental techniques based on, for example, spontaneous infrared fluorescence [6–8], ultraviolet [9] or infrared [10] absorption, and photoionization
[11]. In recent advances, large molecule energy transfer
is being investigated using molecular beam scattering
experiments [12]. From all of these experiments, information has been gathered about the average amount
of energy transferred per collision, including its magnitude, energy dependence (∼linear) and temperature
dependence (∼constant or perhaps ∼linear) [5]. Other
experiments have shown that individual vibrational
modes are activated and deactivated more rapidly than
ENERGY TRANSFER IN MASTER EQUATION SIMULATIONS
others [13,14]. In many systems, the exponential-down
model for the collision step-size distribution (which is
discussed below) provides an adequate description, but
more detailed experiments have shown that a doubleexponential or a “stretched” exponential model gives a
better description [15–17]. In almost all of the studies
cited in this paragraph, the researchers analyzed their
data on the basis of models that assumed that the inelastic collision frequency is independent of the excitation
energy resident in the colliding molecules.
To be clear, unimolecular reactions and master equations can be formulated in terms of the “active energy,”
which, for symmetric tops, consists of the K-rotor and
the vibrations (and internal rotations, if present); because of angular momentum conservation in the unimolecular reactions, the two-dimensional (2D) rotation (of a symmetric top) is omitted from the density
of states calculation in the most common treatments
of angular momentum [1,3]. Since the same density of
states is used in the detailed balance expression for inelastic collisions, the energy transfer parameters refer
to the active energy and not to pure vibrational or rotational transitions. In most cases, the collision frequency
for changing the active energy has been estimated by
assuming Lennard–Jones forces between the collision
partners, but it is well known that the total collision
frequency is much larger [18,19]. In fact, even the collision frequency for pure rotationally inelastic collisions is much larger [20]. It also has long been known
that the rate constant for inelastic energy transfer depends on the energy difference between initial and final
states, as well as on varying propensities for collisions
[14,20–22]. Since the average density of states, which
depends on internal energy, is closely related to the
average energy separation between adjacent states, it
is not surprising that experiments show that the total inelastic collision frequency depends on excitation
energy [23]. Evidence for collision “propensities” suggests mode-specific behavior [14]. However, this sort
of detailed energy transfer information is available for
very few molecular species and is therefore not yet of
general utility. Fortunately, single-channel unimolecular reaction rates depend mostly on the average rate
of energy transfer and are relatively insensitive to the
exact details of the models for collision frequency and
collision step-size distribution [1–3].
Master equation models can, or course, be tailored to individual molecules when the requisite energy transfer data are available. However, to the best of
our knowledge, the complete set of necessary data is
not available for any polyatomic species. In the interest of pragmatism, master equation codes intended for
general use must be flexible enough so that they can be
applied to most polyatomic species and perform with
International Journal of Chemical Kinetics
DOI 10.1002/kin
749
reasonable quantitative accuracy. It is necessary that
they can incorporate the main properties of energy
transfer: the dependence on temperature and internal
energy and the shape of the collision step-size distribution function. Modern codes such as MultiWell [24,25]
succeed in this regard by allowing a wide range of optional choices of collision step-size distribution function and parameterization, including the double exponential and the “stretched” exponential mentioned
above. A second fundamental requirement is that they
accurately simulate the Boltzmann thermal energy distribution, which is needed to predict accurate reaction
rate constants. As shown below, current methods succeed in this regard only at relatively high internal energies: they perform poorly at the low energies that are
important for low threshold chemical reactions (e.g.,
the isomerization of trans-stilbene [26] and the dissociation of prereactive complexes) and for shock tube
simulations.
The purpose of the present paper is to describe a new
approach to the treatment of inelastic collisions that
abandons the physically incorrect assumption that the
inelastic collision frequency is independent of internal
energy and replaces it with an assumption that is qualitatively accurate, although still arbitrary. The resulting
simulations of the Boltzmann distribution are significantly more accurate, and the representation of energy
transfer at low energies is significantly more realistic
than before. At higher energies, the results are quantitatively very similar to the traditional treatment. Because
the new assumption is arbitrary (although qualitatively
reasonable), this approach is only an interim solution.
But breaking with tradition on the inelastic collision
frequency produces a more accurate physical model as
well as a more accurate master equation code.
THEORY
The Master Equation
The collision–reaction master equation [1–3] can be
written using Forst’s simplified notation [1], where
precollision and postcollision energies are written y
and x, respectively:
∞
dW (y, t)
= F (y, t) +
R(y, x) W (x, t) dx
dt
0
∞
−
R(x, y) W (y, t) dx
0
−
channels
i =1
ki (y) W (y, t)
(1)
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BARKER
In Eq. (1), W (y,t) dy is the concentration of a chemical species with vibrational energy in the range y to
y + dy; R(x,y) is the (pseudo-first-order) rate coefficient for collisional energy transfer from initial energy
y to energy x; F (y,t) dy is a source term (e.g., thermal,
chemical, or photo activation); and ki (y) is a unimolecular reaction rate constant for molecules at energy y
reacting via the ith reaction channel. Other terms, such
as those involving radiative emission and absorption
have been omitted.
The rate coefficient R(x,y) is conventionally written as the product of the total vibrationally inelastic
collision frequency kc (y)[M] multiplied by the “collision step-size distribution,” P (x,y), which expresses
the probability density that a molecule initially with
initial energy y, will undergo an inelastic transition to
the energy range x to x + dx:
∞
R(x, y) dx =
R(x, y) dx
0
R(x, y) dx
∞
0 R(x, y) dx
= kc (y)[M]P (x, y) dx
(2a)
∞
P (x, y) dx = 1
W (y)R(x, y) dx = W (x)R(y, x) dy
(4a)
W (y)kc (y)P (x, y) = W (x)kc (x)P (y, x)
(4b)
where Eq. (4b) has been written by using Eq. (2b) and
noting that dx = dy. At equilibrium in the absence of
reaction, the ratio of concentrations is also given by the
ratio of Boltzmann factors:
W (x)
ρ(x)
=
exp[−(x − y)/kB T ]
W (y)
ρ(y)
(5)
where ρ(x) is the density of states at energy x. By
using Eqs. (4b) and (5), we obtain the detailed balance
relationship between the probability densities for upand down-collisions:
kc (x) ρ(x)
P (x, y)
=
exp[−(x − y)/kB T ]
P (y, x)
kc (y) ρ(y)
(6)
(2b)
The first factor on the right-hand side of Eq. (2a), the
integral over the rates of all inelastic transitions from
initial energy y, is the frequency of inelastic collisions,
kc (y)[M], and the second factor (in curly brackets) is
P (x,y) dx. Note that P (x,y) is normalized:
and reverse processes must balance. Therefore,
Collision Step-Size Distributions
The total probability density for an energy changing
collision is normalized (see Eq. (3)) and can be written
as the sum of two integrals corresponding to down- and
up-collisions:
(3)
0
It is important to emphasize that the factorization of
R(x,y) in Eq. (2) is merely for convenience and that
kc (y)[M] and P (x,y) never occur independently of one
another. Furthermore, P (x,y) only has an unambiguous physical interpretation when kc (y)[M] is exactly
equal to the total inelastic collision rate constant. Although neither the exact inelastic collision frequency
nor P (x,y) is known, experimental unimolecular reaction rates and collisional energy transfer data can be
fitted by adopting assumed forms for P (x,y) and kc (y).
According to convention, kc (y) is assumed to be a constant. The inevitable errors in these quantities tend to
compensate for one another when fitting experimental
data and thus it is important to use kc (y) and P (x,y) in
a matched pair whenever possible [2,3,27].
Detailed Balance
By considering detailed balance at equilibrium in the
absence of reactions, the relationship between R(x,y)
and R(y, x) can be found. Detailed balance requires
that in every increment of energy, the rates of forward
y
1 =
∞
P (x, y) dx +
0
P (x, y) dx
(7)
y
To construct a normalized collision step-size distribution (the probability density), it is common practice
to specify a (dimensionless) nonnormalized function
f (x,y), which is assumed to be proportional to P (x,y):
P (x, y) =
f (x, y)
N (y)
(8)
where N (y) is a normalization constant so that P (x,y)
satisfies Eqs. (3) and (7). With this definition, the normalization Eq. (7) becomes
y
1 =
f (x, y)
dx +
N (y)
0
∞
y
f (x, y)
dx
N (y)
(9)
After rearranging Eq. (9), we obtain a formal expression for the normalization constant N (y), expressed as
a sum of integrals:
N (y) =
0
y
f (x, y) dx +
∞
f (x, y) dx
(10)
y
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ENERGY TRANSFER IN MASTER EQUATION SIMULATIONS
or
N (y) = Nd (y) + Nu (y)
(11)
where subscripts d and u denote down-steps (x < y)
and up-steps (x > y), respectively.
After utilizing Eq. (8), the expression for detailed
balance (Eq. (6)) takes the following form:
f (x, y)
N (y) kc (x) ρ(x)
=
exp[−(x − y)/kB T ]
f (y, x)
N(x) kc (y) ρ(y)
(12)
For convenience, the unnormalized function f (x,y) is
usually specified for down-steps, but one could choose
to specify a function for up-steps instead. We will follow convention and specify the function for downsteps, f (x,y) = fd (x,y) with x < y. Thus Nd (y) is
easily evaluated:
y
Nd (y) =
fd (x, y) dx
(13)
0
and Nu (y) can be expressed in terms of fd (x,y) by
using Eq. (12):
∞
Nu (y) =
fd (y, x)
y
N (y) kc (x) ρ(x)
N (x) kc (y) ρ(y)
× exp[−(x − y)/kB T ] dx
(14)
If we had assumed that f (x,y) was specified for upsteps, an analogous procedure would be followed.
Since N (x) appears in the integral expression for
Nu (y), the solution of Eq. (14) is not completely
straightforward. Normalization constant N (y) can be
found by using trial values for N (x) and employing an
iterative solution [28] of Eq. (14), or by rearranging
the equation as follows:
751
inelastic collision rate constants are independent of internal energy: kc (y) = kc (x) = constant [1,3,28,29].
This constant is conventionally identified with kLJ ,
the bimolecular rate constant for collisions between
particles governed by a Lennard–Jones intermolecular potential [1–3]. Experience has shown that kLJ is
a reasonable approximation to the rate constant for
active-energy-changing inelastic collisions involving
large molecules (i.e., with high density of states) with
internal energies much greater than kB T [30–33]. With
this assumption, the ratio of collision rate constants in
Eqs. (12), (14), and (15) is equal to unity. (This conventional assumption was used in all versions of MultiWell
prior to v.2009.0.)
Problems with Normalization
Experience has shown that iterative normalization [28]
converges reasonably rapidly at high energies, but
problems emerge at low energies, where the density
of states is sparse and has large relative fluctuations.
The problems are most severe when an energy grain
that contains just a few states is bracketed on both sides
by energy grains containing much higher densities of
states. Several examples of this behavior are seen in
Fig. 1. For these cases, the normalization factors for
some of the energy grains tend to diverge, instead of
converging during the iterative calculation. Because of
this problem, it has been necessary to limit the number of iterations to, e.g., 2–5, so that normalization
at high energy converges sufficiently, while normalization at low energy does not diverge too much. In
previous work, more iterations were sometimes used
to achieve better performance in certain energy ranges
[34]. As the number of iterations is increased, results
at higher energies systematically become more accurate, while those at low energies become less so. If the
y
N (y) =
1−
∞
y
fd (x, y) dx
fd (y, x) kc (x) ρ(x)
exp[−(x − y)/kB T ] dx
N (x) kc (y) ρ(y)
0
Equation (15) can be solved with the finite difference
algorithm described by Gilbert and coworkers [2,29].
Both of these approaches to finding N(y) are based on
specifying fd (x,y) and requiring that N (y) first be estimated at very high energies, well above the energies
of interest, where Nd (y) and Nu (y) tend to become independent of energy (at least when the average energy
transferred per collision is independent of energy).
In general, kc (y) is expected to depend on the initial
energy, y, but it is common practice to assume that the
International Journal of Chemical Kinetics
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(15)
number of iterations is not constrained, the normalization factors calculated for some individual energy
grains at low energies appear to approach ∞ and
can experience numerical overflows. The strategy of
limiting the number of iterations, although not completely satisfactory, is reasonably effective in producing steady-state energy distributions that simulate the
equilibrium Boltzmann distribution; and example of
the results obtained in this way is shown later in Fig. 3.
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BARKER
Figure 1 Normalization constants (lower panel) computed using the method of Gilbert and coworkers [2,29], the conventional
(iterative) approach [40], and the new approach. Densities of states are shown in the upper panel. Note that the energy grain
at 100 cm−1 contains no states and one of the negative excursions (clipped at the lower plot boundary) extends to almost
−3000 cm−1 . At higher energies the three normalization approaches become indistinguishable. [Color figure can be viewed in
the online issue, which is available at www.interscience.wiley.com.]
Related problems arise at low energies when using
Eq. (15) and the finite difference algorithm of Gilbert
and coworkers [2,29]. The Gilbert algorithm is quite
general, but in practice the pragmatic assumption is
made that the collision frequency is independent of
excitation energy. The algorithm is very accurate at
higher energies, but it is not convenient for use with a
continuous master equation such as MultiWell, which
interpolates between widely separated energies at high
excitation energies and adopts a contiguous set of energy grains at low energies. In addition, the Gilbert algorithm does not assign a value for N (y = 0). Aside
from these considerations, the Gilbert algorithm does
not perform well at low energies for the standard
exponential-down model when assuming that kc (y)
is independent of energy (i.e., the conventional approach). An example of this poor behavior is shown
in Fig. 1, where the excursions of the normalization
constant are not only large (clipped at the boundary
of the plot), but negative and therefore unphysical. It
is also significant that the negative excursions occur
at the same energies as the large positive excursions
obtained using the iterative method described above.
For most master equation simulations, the lowest
relevant energies are a few kB T below the reaction
threshold energy, which is high enough so that the normalization problems do not emerge. However, in shock
tube simulations [34–36] and in simulating reactions
with very low energy barriers [26] the relevant energies extend all of the way to the bottom of the energy
ladder. Furthermore, in 2D (E,J ) master equation simulations, the normalization problems are exacerbated
because the 2D density of states is even more sparse
and hence exhibits even greater fluctuations than in the
one-dimensional (1D) version.
One solution to these problems is to forego the use
of direct counts of states and to instead use smooth
approximations such as the Whitten–Rabinovitch approximation [37] or the method of steepest descents
[1]. Forst has pointed out that smooth approximations
are more appropriate for computing microcanonical
rate constants, k(E), from RRKM theory, because the
theory is based on semiclassical approximations [1].
However, a fine-grained density of states function is
needed to achieve accurate agreement with the quantum Boltzmann distribution when nonreactive energy
transfer is of key importance, as in shock tube simulations and in the reactions of weakly bound complexes.
For that reason, we prefer to use methods based on
exact counts of states whenever possible.
One possible explanation of the large positive (or
negative) excursions is sensitivity to numerical noise.
This possibility is apparent in Eq. (15). At y = 0,
the zero-point energy, both the numerator and the denominator in Eq. (15) are identically equal to zero.
At low energies above zero, the denominator is small
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ENERGY TRANSFER IN MASTER EQUATION SIMULATIONS
and the expression is sensitive to numerical errors. As
the energy grain size is made smaller, the sensitivity
becomes larger. However, numerical noise is minimal
when using a double-precision version of the Gilbert algorithm, and it is not the only cause of the large negative
excursions.
Because the Gilbert algorithm is based on very general principles [2,29] and still produces unphysical results suggests that there is a fundamental deficiency in
the conventional model for collisional energy transfer.
The only feature of the conventional model that is not
required by the Gilbert algorithm is the conventional
assumption that the inelastic collision rate constant
kc (y) is independent of energy, y. Thus we identify
this assumption with the unphysical results. In the next
section, we argue that the inelastic collision frequency
should vary significantly with energy, especially at low
energies. A simple revision to the conventional approach is then proposed, which eliminates the numerical problems with normalization, yields an energydependent inelastic collision frequency, and produces
more accurate simulations of the Boltzmann distribution at low energies. Although this revision is arbitrary,
it is at least physically plausible and behaves in a qualitatively correct manner.
REVISIONS TO THE NORMALIZATION
ALGORITHM
As pointed out above, kc (y) is expected to depend on
internal energy, y. Both experiments and theories of
state-to-state inelastic energy transfer show that the
rate of inelastic collisional energy transfer is very slow
when adjacent states are widely separated in energy,
but the rate becomes very fast when state densities are
high [3,5,13,20]. Put another way, the rate of inelastic
collisions is very small in the state-to-state region at
low energy and rather large at high energy where the
density of states is high [20]. In standard master equation treatments, elastic collisions are ignored, since
they add to the computational burden and do not result
in energy changes. Thus the rate constant for inelastic
collisions is the relevant quantity for master equation
simulations, and it tends to increase as the density of
states increases, at least at low energies.
The normalization factor N(y) also tends to increase
with energy. This is because the integral over downsteps increases with energy (at least at low energies),
whereas the integral over up-steps remains more or
less constant or increases with energy as the number
of empty energy grains decreases. Typically, N (y) increases from a low value at y = 0 and becomes constant within a few percent when the density of states
International Journal of Chemical Kinetics
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753
reaches ∼100 states/cm−1 (for 10 cm−1 energy grains
and an exponential model with constant energy transfer parameter α; see below). This behavior is also seen
in a statistical dynamical model energy transfer model
published a few years ago [38,39].
We argue that in principle it should be possible to
express kc (y) as a function of N (y), but the specific
functional dependence is not known. In the absence
of specific knowledge about kc (y), we make a simple, pragmatic assumption that kc (y) is directly proportional to N (y). With this assumption, the following
ratio, which appears in Eq. (14), equals unity:
N (y) kc (x)
= 1
N (x) kc (y)
(16)
and Eq. (14) becomes
∞
Nu (y) =
fd (y, x)
y
ρ(x)
exp[−(x − y)/kB T ] dx
ρ(y)
(17)
This expression, like Eq. (13), requires only straightforward integration. For the purpose of numerical integration, the integral can be converted to a sum and
Eq. (17) becomes
Nu (j ) = Egrain
jmax
i = j +1
fd (j, i)
ρ(i)
ρ(j )
× exp[−(i − j )Egrain /kB T ]
(18)
where i and j are index numbers, jmax is the highest
energy grain in the model, and Egrain is the energy
grain size. Because the normalization constant does
not exist if there are no states, empty energy grains are
omitted when evaluating Eq. (18). Thus, ρ(j ) is always
>0, no singularities are possible, and the normalization
constant N (y) = Nd (y) + Nu (y) is greater than zero
and absolutely stable. No iterative methods or finite
difference algorithms (i.e., the Gilbert algorithm) are
needed for its evaluation. No smoothing of the density
of states is required. As a result, the normalization at
low energies is never less than zero and it does not show
excursions that are as extreme as those found using the
conventional (iterative) method, as shown in Fig. 1. In
the calculations carried out below, the trapezoidal rule
was used instead of the simple sum shown in Eq. (18),
but the considerations are the same.
The assumption that kc (y) is directly proportional
to N (y) requires that the collision frequency for
a molecule with excitation energy y be calculated
using kc (y) = CN (y) (see Eq. (16)), where the
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BARKER
proportionality constant C must be established by some
other means. Since the Lennard–Jones rate constant
kLJ conventionally has been used for calculating the
low-pressure limit of unimolecular and recombination
reactions, we have adopted the following form for the
total energy-dependent rate constant for the inelastic
collision frequency:
kc , as discussed in the Introduction. This is especially
true for 2D master equation models, in which rotational
energy transfer is considered explicitly, since the rate
constants for rotationally inelastic collisions are much
larger than kLJ .
COMPUTATIONAL METHODS
kLJ
N (y)
kc (y) =
N(Eref )
(19)
where Eref is a reference energy. By combining Eq. (19)
with Eqs. (16), (12), (8), and (2) and incorporating the
definition of fd (x,y), we obtain the following expressions for the rates of energy transfer:
Down-steps:
for x < y
Rd (x, y) dx = kLJ [M]
fd (x, y) dx
N (Eref )
(20a)
Up-steps: for x < y
Ru (x, y) dx = kLJ [M]
ρ(x)
ρ(y)
× exp[−(x − y)/kB T ]
fd (y, x) dx
N(Eref )
(20b)
These expressions are easy to implement since the
normalization constant need only be computed at the
reference energy Eref . Moreover, since the rates of
collisional energy transfer do not depend on N (y),
the energy-dependent normalization factor, the population distributions are not affected by errors in
normalization.
Although the energy-dependent normalization factor N(y) does not appear in Eq. (20), it is important to
examine its behavior. In most cases, N (y) is relatively
smooth at energies where the density of states is ≥100
states/cm−1 ; this condition provides a good criterion
for specifying Eref . Other choices for Eref will produce
numerically different collision frequencies and energy
transfer parameters, but above the energy where the
density of states becomes relatively smooth, there is
little further difference (see Fig. 1), unless the energy
transfer parameter α depends on energy. In reaction
studies, the rate of energy transfer is most important
at energies near the reaction critical energy. For nonreactive systems, the density of states criterion may be
used (see Appendix A for the criteria used in the latest
version of MultiWell). As discussed below, the numerical results obtained using this new approach are nearly
the same as those obtained using the old conventional
approach, except at low energies. However, it should
be noted that kLJ may not always be the best choice for
It should be emphasized that the new approach is quite
generally valid for all energy-grained master equation
formulations, whether solved by a stochastic method or
by matrix diagonalization. Care must be taken so that
energy grains that do not contain any states also never
contain any population. Such “empty” energy grains
can be omitted from the calculation. For energy grains
that do contain states, Eq. (18) provides the basis for a
very simple algorithm for computing the normalization
factor N (y), since it does not depend on the N (x) for
any other energy grain. The Gilbert algorithm is not
required: Eq. (18) requires only a simple numerical
integration. Of course, the collision rate constant kc (y)
varies with energy y, as described by Eq. (19).
The present calculations were carried out using versions of the MultiWell Program Suite [24]. The conventional calculations were carried out using version
2008.3, which consists of the free-source code, examples, and MultiWell User Manual that are published as
a package on the Internet [40]. Note that version 2008.3
utilizes three (3) iterations when calculating the “conventional” normalization constant, as discussed above.
The calculations carried out using the new approach
(described in the preceding section) are not iterative
and were performed with a development version of
MultiWell (which is now available as v.2009 on the
MultiWell Web site [25]).
The MultiWell master equation is based on a “hybrid” approach, which consists of a continuum master
equation formulation at high energies and an energygrained master equation at low energies. Densities of
states and other quantities are stored in “double arrays,” where the energy grain at low energy is small
enough (typically 5 or 10 cm−1 ) for proper numerical convergence; much larger energy steps (typically
≥100 cm−1 ) are stored for higher energies. The dividing line between the upper and lower energy regimes
is typically set at the lowest energy where the grain-tograin difference in densities of states is less than a few
percent and the density of states function is relatively
smooth. The numerical integrations were carried out by
using the trapezoidal rule in the lower energy regime.
In the upper energy regime, the integrations were carried out analytically for each energy step by noting that
the density of states is an approximately exponential
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ENERGY TRANSFER IN MASTER EQUATION SIMULATIONS
function of internal energy over the limited range of
each energy step.
Once again, it should be emphasized that the new
approach to normalization is appropriate for all master
equation codes, not just MultiWell.
Densities of states were calculated by exact counts
using “Densum,” one of the codes included in the
MultiWell Program Suite. Densum employs the Beyer–
Swinehart algorithm [41] as implemented by Stein and
Rabinovitch [42] to calculate sums and densities of
states based on any combination of separable degrees
of freedom consisting of molecular harmonic oscillators, anharmonic oscillators, free internal rotations,
hindered internal rotations, and particle-in-a-box. The
vibrational frequencies, moments of inertia, and other
parameters used for all of the present calculations are
summarized in Appendix B.
PERFORMANCE TESTS AND RESULTS
The standard exponential-down model of energy transfer, which is used in most master equation simulations,
is used for the performance tests:
−(y − x)
(21)
fd (x, y) = exp
α(y)
where α(y) is a polynomial [5]. In most cases only the
constant term of the polynomial is needed to obtain
satisfactory fits to unimolecular or recombination reaction rate data, but the linear polynomial coefficient is
also needed to accurately fit energy transfer and shock
tube data. By adjusting these coefficients, it is possible
to compensate, at least in part, for possible errors in the
collision frequency (assumed to be given by kLJ [M])
and in other elements of the master equation model. It
should be emphasized that other energy transfer model
functions are more accurate than Eq. (21) for the relatively few cases where the parameters are known [5],
as discussed in the Introduction. Many other models
(including the double exponential and the stretched exponential model [15–17,43]) have been investigated by
assorted research groups and most of those functions
can be selected in MultiWell. In most cases, however,
energy transfer parameters are known very poorly, if at
all, and thus the Eq. (21) with constant α is the model
used by most researchers. Thus it is adopted for the
tests presented here.
In this context, it is appropriate to test the accuracy
of the present new approach and to determine whether
it is still possible to fit experimental data satisfactorily
by varying the polynomial coefficients in α(y). Four
types of tests have been performed, as described below,
with both the old and new approaches.
International Journal of Chemical Kinetics
DOI 10.1002/kin
755
Boltzmann Distribution
If the master equation is accurate, simulations for
sequences of multiple collisions should converge accurately to the Boltzmann distribution in a thermal
system as the number of collisions is increased indefinitely. Expressed in terms of densities of states, the
Boltzmann distribution is
B(y; T ) =
1
ρ(y) exp[−y/kB T ]
Q(T )
where the partition function Q(T) is given by
∞
Q(T ) =
ρ(y) exp[−y/kB T ] dy
(22)
(23)
0
The average thermal energy is given by
∞
ET =
yB(y; T ) dy
(24)
0
In each of the first series of tests, master equation simulations were carried out for 1000 random sequences
of 1000 collisions each. The initial energy distribution was selected by Monte Carlo techniques from
the Boltzmann distribution function. The selection of
the initial energy distribution and computation of yT
are standard features in the MultiWell master equation code [24,40], where the integrals in Eqs. (23) and
(24) are evaluated numerically with the trapezoidal
rule. After every collision, the energy was recorded.
This procedure produced 106 values per test, which
were averaged to obtain the average thermal energy
yT ,test . This value can then be compared to ET
obtained independently by numerical integration of
Eq. (24). Tests were carried out by using two constant values of the energy transfer parameter α(y) for
ClOOCl (small molecule, colliding with N2 ) and for
1,1,1-trifluoroethane (intermediate size, colliding with
Kr). Tests carried out for norbornene (large molecule,
colliding with Kr) employed an energy-dependent parameter α(y) found from an analysis [36] of shock tube
experimental data [44].
The results for ClOOCl and 1,1,1-trifluoroethane
are given in Table I, and those for norbornene are given
in Table II. In the tables, ET is the average thermal
energy obtained by numerical integration of Eq. (24), α
is the energy transfer parameter, Enew is the average
thermal energy obtained using master equation simulations that incorporate the new normalization method,
and Econv is the average thermal energy obtained
using master equation simulations that incorporate the
conventional normalization method. MultiWell utilizes
the stochastic simulation algorithm, which has associated errors for a finite number of samples [45,46]. The
756
BARKER
Table I Tests of the Simulated Average Thermal Energy
ET (cm−1 )
T (K)
200
200
400
400
800
800
231
231
825
825
2413
2413
200
200
400
400
800
800
1600
1600
2400
2400
202
202
948
948
3683
3683
11,480
11,480
20,482
20,482
202
202
948
948
3683
3683
200
200
400
400
800
800
α (cm−1 )
Econv (cm−1 )
ClOOCla
500
100
500
100
500
100
Enew (cm−1 )
259
226
734
—
2671
2261
230
245
831
—
2413
2446
1,1,1-Trifluoroethaneb
500
100
500
100
500
100
500
100
500
100
218
189
874
738
3589
3458
11,474
11,364
20,336
20,302
228
240
973
996
3691
3690
11,507
11,472
20,547
20,458
1,1,1-Trifluoroethanec
500
100
500
100
500
100
222
180
830
691
3589
3410
222
230
970
991
3713
3758
Egrain = 5 cm−1 .
Egrain = 10 cm−1 .
c
Egrain = 5 cm−1 ; sampling errors in Econv and Enew are ≤1%.
a
b
stochastic sampling errors in Econv and Enew presented in Tables I and II are ≤1%.
Inspection of Tables I and II shows Econv often differs significantly from ET . This is because the three
iterations used in the iterative normalization scheme in
MultiWell-v.2008.3 are not sufficient for complete numerical convergence. The use of just three iterations is
a compromise between achieving reasonable accuracy
at higher energies and increasingly inaccurate results
at lower energies, as discussed above. The computed
Table II Norbornenea
T (K) ET (cm−1 ) Econv (cm−1 ) Enew (cm−1 )
200
400
800
1200
a
207
1476
7679
16,602
185
1231
7206
15,555
255
1531
7642
16,569
Egrain = 10 cm−1 ; α(E)/cm−1 = 40 + 0.0063 × E.
thermal energy distributions are constrained by both
detailed balance and normalization. The computed average thermal energies are different because the conventional method is affected by the problems with normalization and the new method is not. Since the errors
in normalization constant N (y) are energy dependent,
the energy distributions are in error, resulting in the
discrepancies in the average thermal energy ET .
The tables also show that the new approach is more
accurate than the conventional one for most combinations of parameters. The following observations come
from inspection of the tables:
•
•
•
Enew usually tends to be slightly greater than
ET .
Larger values of α tend to give slightly more accurate results for Enew , but not necessarily for
Econv .
Smaller values of Egrain tend to improve the
accuracy of Enew at lower temperatures, but that
is not always the case at higher temperatures.
International Journal of Chemical Kinetics
DOI 10.1002/kin
ENERGY TRANSFER IN MASTER EQUATION SIMULATIONS
•
At higher temperatures, Enew is uniformly more
accurate than Econv , but that is often not the case
at low temperatures.
Since the average thermal energy is a function of
temperature, it is interesting to convert the energy
errors to temperature differences. For example, at
1200 K Econv for norbornene is in error by −1047
cm−1 . This corresponds to a temperature difference
T = −43 K, or a relative temperature error of
−3.6%. In contrast, the error in Enew at the same
temperature is only −33 cm−1 and Tnew = −0.2%.
This error is so small that it may just be due to
stochastic sampling error, rather than to an intrinsic
deficiency in the new approach.
In the second series of tests at two temperatures, 100
samples were taken from each of 104 random collision
sequences, binned in 10 cm−1 energy grains, normalized, and then displayed for comparison with the exact
Boltzmann distribution calculated using Eq. (22). Examples of these tests are shown in Figs. 2 and 3. At both
temperatures, the energy distributions computed using
the new approach are in excellent agreement with the
Boltzmann distribution. Not only are the gross features
correct, but most of the detailed structure is reproduced
accurately. This performance contrasts with that of the
conventional approach (see Fig. 3), which is unable
to simulate the rapid oscillations at low energies and
757
misses systematically at higher energies. Note, however, that conventional method simulates the shape of
the energy distribution reasonably well at the higher
energies.
Energy Transfer Decay
In this set of tests, simulations were carried out by monitoring the average energy of an ensemble of molecules
initially excited to a relatively high energy. Energy
transfer was assumed to be governed by an exponentialdown model (Eq. (21)) with energy transfer parameter
α that was independent of energy.
The total energy transferred per collision, Eall
can be written as the sum of the average energies transferred in activating and in deactivating collisions:
E(y)all =
0
y
(x − y) P (x, y) dx
∞
+
(x − y) P (x, y) dx (25a)
y
= −Edown + Eup
(25b)
where Eup is for activation and Edown is for
deactivation. For an exponential model with energy
transfer parameter α that is independent of energy, it
has been shown that Eall can be estimated by using
Figure 2 Sampled collision energies from the new method vs. the exact Boltzmann energy distribution at 200 K. Both the
thermal energy distribution and the sampled energies were binned in 10 cm−1 energy grains. Statistical fluctuations in the
sampled energies are apparent at the highest energies shown. The average thermal energies differ by only ∼21 cm−1 . Most of
the detailed energy structure in the exact distribution is reproduced in the sampled distribution. [Color figure can be viewed in
the online issue, which is available at www.interscience.wiley.com.]
International Journal of Chemical Kinetics
DOI 10.1002/kin
758
BARKER
Figure 3 Sampled collision energies vs. exact thermal energy distribution at 500 K:106 energy samples (100 samples in each
of 104 stochastic trials). The thermal energy distribution is shown with 10 cm−1 energy grains, and the sampled energies were
binned in 30 cm−1 energy grains. The average thermal energy obtained using the new method Enew is only ∼15 cm−1 different
from the exact value. The old, conventional method gives Econv in error by ∼100 cm−1 . Most of the detailed energy structure
in the exact distribution is reproduced using the new method, but not by the conventional method. [Color figure can be viewed
in the online issue, which is available at www.interscience.wiley.com.]
the following equation [47]:
2 E(y)all ≈ [1/kB T + 1/α − B]−1 − α (25c)
where B = d(ln(ρ))/dy. From Eq. (25), it is apparent
that Eall is nearly constant as long as y > α.
When Eall is nearly constant, the resulting decay
of average energy is expected to be a nearly linear function of time. This expectation is confirmed by the simulations shown in Fig. 4. The initial energy decay rate
obtained using the conventional approach is 3.0 × 108
cm−1 s−1 , and at the simulated pressure of 0.1 Torr the
collision frequency is 2.9 × 106 s−1 . Together, these
values give Eall = 103 cm−1 , using the conventional approach. The estimate from Eq. (25c) is in good
agreement with this value: Eall,24c = 109 cm−1 .
This estimate is based on α = 200 cm−1 , T =
300 K, and B = 6.41 × 10−4 (cm−1 )−1 , which was
found from a least-squares fit of the toluene density of
states from 35,000 to 40,000 cm−1 .
The initial energy decay rate obtained using the new
approach is 4.6% slower than the simulation that used
the conventional approach. This minor difference is
mostly due to the difference in collision frequencies.
As discussed above, the new approach adopts kLJ at a
reference energy and assumes the collision rate constant kc is proportional to the normalization constant.
At the internal energy of 40,000 cm−1 , the normal-
ization constant is 5.6% smaller than at the reference
energy (4990 cm−1 in these simulations), resulting in a
slower collision frequency and hence a slower energy
decay rate. After correcting for the change in collision
frequency, the simulation with the new approach gives
Eall = 105 cm−1 , in very good agreement with the
conventional method, as expected for relatively high
energies when α is a constant.
Figure 4 shows that simulations carried out with
constant α are in very good agreement over the entire energy decay. The simulations carried out using the energy-dependent α (Fig. 5) also show very
good agreement. Here the simulations are more realistic, since the energy transfer parameter α(y) is
the energy-dependent function determined in kinetically controlled selective ionization (KCSI) experiments [48]. Most of the difference between the two
approaches arises from minor differences in collision
frequency. It is clear that experimental data can be
fitted equally well using either method, and the fitted
parameters will differ because of the energy-dependent
inelastic collision rate coefficient.
Unimolecular Reactions in Shock Waves
In shock wave experiments, the reactant gas mixture
consists of a buffer gas and a few percent or less of a
reactant gas. The gas mixture is subjected to a shock
wave, which adiabatically compresses the gas mixture
International Journal of Chemical Kinetics
DOI 10.1002/kin
ENERGY TRANSFER IN MASTER EQUATION SIMULATIONS
759
Figure 4 Simulated deactivation of excited toluene in 0.1 Torr of argon at 300 K, assuming the energy transfer parameter α = 200 cm−1 (independent of energy). [Color figure can be viewed in the online issue, which is available at
www.interscience.wiley.com.]
and raises its translational temperature on a submicrosecond time scale [49]. Thus the reactant, initially
at thermal equilibrium with the bath at ambient temperature, finds itself immersed in a bath at much higher
temperature. The reactant, initially with a vibrational
energy distribution that corresponds to the initial ambient temperature, undergoes collisions until it either
decomposes or its energy distribution reaches equilibrium with the new translational temperature.
In the shock tube experiments of Kiefer et at. [44],
the reactant was norbornene and the bath gas was krypton. The experiments were later modeled successfully
by Barker and King [36], who used a master equation
that was similar in many respects to MultiWell.
To accurately model shock tube experiments is
a very demanding task for master equation simulations. Not only must the chemical reaction be modeled accurately, but the whole evolution of the energy
Figure 5 Simulated deactivation of excited toluene in 0.1 Torr of argon at 300 K, assuming the energy transfer parameter α(y) obtained in KCSI experiments [48]. [Color figure can be viewed in the online issue, which is available at
www.interscience.wiley.com.]
International Journal of Chemical Kinetics
DOI 10.1002/kin
760
BARKER
distribution must be modeled accurately as it evolves
from the initial low-temperature thermal distribution
to the final steady-state (“fall-off”) distribution that is
the result of competition between collisional energy
transfer and chemical reaction. Thus the master equation model must be accurate essentially all of the way
from near the reactant zero-point energy to high above
the reaction threshold energy.
Experimental knowledge about the evolving energy
distribution comes from several kinds of information
[36,44]. These include the pressure- and temperaturedependent unimolecular reaction rate constants (kuni )
measured after the energy distribution has established
the final steady state. But more direct dynamical information on energy transfer comes from the measurement of the rate of collisional energy transfer that accompanies the evolution of the energy distribution and
the closely related incubation time (τinc ). The reaction
incubation time is defined as the time delay from the
arrival of the incident shock wave to the onset of reaction. This time delay arises because of the time needed
to collisionally activate the molecule to energies above
the reaction threshold.
In the tests performed here, the model parameters
found in our previous work [36] are used with the
conventional and new approaches to ascertain the impact of the new approach. As discussed above, the two
approaches are expected to give somewhat different
results. Since master equation models are only approximations to the true physico-chemical system, we are
not in a position to determine which model is more
correct. Thus the only question to be addressed here is,
can the new approach be used successfully to describe
the experimental data?
To answer that question, results obtained using
the conventional and new approaches are first compared, and then new calculations are carried out using the new approach but with revised parameters
to determine whether the new approach can accurately model the results obtained using the conventional one with the original parameters. The energy
transfer model chosen for present purposes is Model
#3 from Barker and King [36]: the exponential-down
model with α(y) = 40 + 0.0063y. The specific experiment that is used for illustration is Shock #76 reported
by Kiefer et al. [44] and previously modeled in Barker
and King [36].
The calculated fraction of norbornene that survives
is plotted as a function of time in Fig. 6 for both the conventional and new approaches. Qualitatively, the two
approaches are similar to the previous model calculations carried out by Barker and King [36], but clearly
the two approaches differ quantitatively as expected
for an energy-dependent α(y). To extract quantitative
Figure 6 Simulation of norbornene decomposition in shock
waves. The conditions correspond to Shock #76 reported by
Kiefer et al. [44] and modeled previously by Barker and King
[36]. [Color figure can be viewed in the online issue, which
is available at www.interscience.wiley.com.]
estimates of the steady-state unimolecular rate constant
kuni and incubation time τinc , it is convenient to fit the
data in Fig. 6 by using the following function:
N (t)
= exp [−kuni (t − τinc )][1 − exp(−ct)]b (26)
N (0)
where parameters c and b (which typically take values
of ∼1000 and 0.2, respectively) empirically describe
the time evolution of kuni from zero to its final steadystate value. This is the correct form of Eq. (9) in Barker
and King [36] (which contained typographical errors).
From least-squares fits of the data in Fig. 6, the
steady-state values for kuni are 2.69 × 104 s−1 and
3.56 × 104 s−1 (at 1197 K and 48.3 Torr of Kr bath
gas) for the conventional and the new approach, respectively; the corresponding values for τinc are 7.8 and
10.2 μs, respectively. The differences between these
two results are due to several factors. First, the new approach gives a better description of the thermal energy
distribution, resulting in differences in the computed
steady-state unimolecular rate constants. Second, because the energy transfer parameter α(y) is a function
of energy, the normalization constant is a function of
energy, even in the upper energy regime. Therefore,
the constant, energy-independent collision frequency
in the conventional approach is replaced by an energydependent function, resulting in differences in the vibrational relaxation time and incubation time.
The key question is whether or not the new approach
can be empirically parameterized to fit the experimental data. Since the data were originally modeled
by using the conventional approach, we assume the
conventional model is a reasonable description of the
International Journal of Chemical Kinetics
DOI 10.1002/kin
ENERGY TRANSFER IN MASTER EQUATION SIMULATIONS
experimental data. By varying parameters, we found
that by using α(y) = 70 + 0.003y, the new approach
reproduces the results obtained with the conventional
method, with deviations smaller than the width of the
heavy dashed line in Fig. 6. This result shows that the
new approach is just as effective at fitting experimental data as the conventional one, even for shock tube
experiments, which are highly sensitive to the accurate
treatment of collisional energy transfer. The numerical
values of the parameters are different, but this is mostly
because the inelastic collision frequency is no longer
assumed to be independent of energy.
CONCLUSIONS
The ultimate goal of this work is to improve the accuracy of master equation simulations. The specific aims
of this paper were to introduce a plausible new physical assumption about the inelastic collision frequency,
to determine whether the new approach is a significant
improvement over the old, and to ascertain whether the
new approach will still allow one to fit experimental
data empirically. A new assumption is needed because
the old assumption that the rate constant for inelastic
collisions kc (y) is independent of energy, y, results in
normalization constants, N (y), that are nonphysical at
low energies, or ill-behaved numerically.
The plausible new assumption is that the rate constant for inelastic collisions kc (y) is directly proportional to the normalization factor for the collision stepsize distribution N (y). This assumption is qualitatively
correct, since both kc (y) and N (y) depend on energy, y.
However, little is known about kc (y). For that reason,
we arbitrarily postulate a direct proportionality, but this
point will have to be addressed again in the future. To
use parameters that have magnitudes that are similar
to previous conventional master equation simulations,
we define a reference energy, Eref , at which kc (Eref ) is
assumed to equal the rate constant for Lennard–Jones
collisions, kLJ .
Simulations based on the new approach show that it
eliminates the nonphysical and ill-behaved normalization constants, which motivated this work. They also
show that the new approach does a significantly better job over-all in simulating the Boltzmann distribution function. Simulations with empirically adjusted
parameters were capable of fitting the previous conventional simulations of several experiments, showing
that the new approach is capable of fitting experimental
data.
On the basis of these considerations, we have
adopted this new approach in our own master equation
simulations. We expect this new approach will also
International Journal of Chemical Kinetics
DOI 10.1002/kin
761
be useful in 2D (E,J resolved) master equation implementations, where the numerical ill-behavior may
be even worse than in the 1D master equation studied here. The new approach performs better than the
conventional model, but it must eventually be replaced
when knowledge of the total inelastic collision frequency is improved. That, however, may not happen
for some time to come.
Thanks go to Jingyao Liu (Jilin University) for providing calculated anharmonicity coefficients and the hindered internal
rotation barrier for ClOOCl, to Yi Jie Chua, an undergraduate
researcher here at Michigan, for assistance with some preliminary calculations, and to Andrea Maranzana (University
of Torino) for useful discussions.
APPENDIX A: CRITERIA IN MULTIWELL
FOR E r e f
For a description of double arrays and for definitions
of terms, see the MultiWell User Manual [25].
In MultiWell, we identify Eref with the critical energy of the lowest reaction threshold energy (when
multiple reaction channels are involved) that is higher
in energy than the energy boundary (parameter Emax1)
between the lower and upper portions of the double array. If no reaction threshold energies are below
Emax1, we arbitrarily specify Eref as equal to Emax1.
At Emax1, N (y) is a relatively smooth function and
the density of states is typically >10–100 states/cm−1 .
APPENDIX B: PARAMETERS FOR THE
MASTER EQUATION SIMULATIONS
Each excited species was treated approximately as a
symmetrical top with a 1D K-rotor and a 2D adiabatic
rotation. All densities of states were calculated using
the “active energy,” which consists of the K-rotor and
the vibrations; because of angular momentum conservation in the unimolecular reactions, the 2D rotation
is omitted from the density of states calculation in the
most common treatments of angular momentum [1,3].
Since the same density of states is used in the detailed
balance expression for inelastic collisions, the energy
transfer parameters refer to the active energy and not
to pure vibrational or rotational transitions.
Vibrational frequencies for ClOOCl were taken
from Jacobs et al. [50] with vibrational anharmonicities
calculated using density functional theory (B3LYP/6311+G(3df)) [Jingyao Liu, private communication,
2008]. The lowest vibrational frequency (127 cm−1 )
762
BARKER
Table AI ClOOCl Vibrational Frequencies
ω◦
(cm−1 )
ωe Xe
−9.329
−3.061
3.889
0.069
−1.695
Hindered rotor barrier = 5493 cm−1a
754
648.
543
419.
321.
127.
46.32 amu Å2
a
Moment of inertia
(K-rotor)
Twofold internal rotation.
Table AII Lennard–Jones Parameters
Species
σ (Å)
ε/kB (K)
Reference
N2
Kr
ClOOCl
1,1,1-CF3 CH3
Norbornene
3.74
3.61
4.84
4.959
5.5
82
190
367
387
330
[51]
[52]
Estimated
[52]
[44]
was treated as a twofold hindered internal rotation
with hindrance barrier height 5493 cm−1 [Jingyao Liu,
private communication, 2008]. These parameters are
summarized in Table AI.
Vibrational frequencies, moments of inertia, and
other parameters can be found in previous papers
on 1,1,1-trifluoroethane [34] and norbornene [36].
Lennard–Jones parameters for all of the collision partners are summarized in Table AII.
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